Circle is similar to a polygon with infinite number of sides It is known from the time of Euclid, that a circle is similar to a polygon with infinite number of sides.
But this ^^ is informal.
Do you know any formalization where it appears that a circle is a polygon with infinite number of sides?
 A: The idea of viewing a circle as a (regular?) infinite-sided polygon goes back at least as far as Nicholas of Cusa (sometimes referred to as Cusanus).  The idea was picked up by Kepler who used it in area calculations, many years before the idea was developed formally in a mathematically adequate form. The torch was picked up by Leibniz who most likely introduced the term infinitesimal so that the polygon now becomes infinitesimal-sided.
In the 18th century, Leibniz's ideas were developed by his followers like Johann Bernoulli, and his followers' followers like Leonhard Euler, who used both infinitesimals and infinite numbers in a routine way.
In the 19th century, infinitesimals were still in common use chez Cauchy, who used them to define continuity of a function $f$ as follows: $f$ is continuous if each infinitesimal increment $\alpha$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function.
The next generation of mathematicians developed set-theoretic foundations that ultimately formalized the real numbers, but failed to formalize infinitesimal procedures of the founders of the calculus.
The work on infinitesimals continued throughout the second half of the 19th century and the beginning of the 20th century, by people like Paul du Bois-Raymond, Veronese, Hahn, Dehn, Hilbert, and others. Skolem in 1933 introduced a model of Peano arithmetic containing infinite numbers.
It was not until the 1960s that Abraham Robinson pulled all of these strings together by creating a modern framework for working with infinitesimal and infinite numbers that meets current standards of mathematical rigor.
In Robinson's framework, one can approximate the circle by a regular polygon with $H$ sides where $H$ is an infinite hypernatural number.  More precisely, the circle is the standard part of the infinite-sided polygon.
